Let
us imagine two tone-generating points surrounded by circles of equidistant
waves. At some point, depending on the distance between the points, these
circles of waves will intersect. In reality, of course, they will always be
spheres, but projection on a plane is sufficient to discover the laws by which these
intersection points occur. One must then simply imagine the relevant figures
transposed into the spatial realm, turning an ellipse into an ellipsoid, a
parabola into a paraboloid, and a hyperbola into a hyperboloid.

If
we connect the intersection points of the two groups of concentric circles, we
will trace out ellipses or hyperbolas (Fig. 167), depending in which direction
we proceed. Since Fig. 167 is very easy to draw, the reader can derive the
formula of the ellipse (Fig. 167a) and the hyperbola (Fig. 167b) by counting
off the radii that generate the respective intersection points. This shows that
the ellipse traces all the points for which the sum of their distance from A
and their distance from B is equal, while the hyperbola traces all the points
for which the distance from A minus the distance from B is equal.

Figure 167a

Figure 167b

Figure 168

We
have thus achieved the derivation of the ellipse and the hyperbola through the
intersection of two fundamental phenomena of general vibration theory: the two
wave-spheres.

We
can read off the parabola directly from our diagram (Fig. 168). Its ratios are:

Major

0/2

1/1

(0/0)

0/3

2/2

2/1

(0/0)

0/4

3/3

4/2

3/1

(0/0)

0/5

4/4

6/3

6/2

4/1

(0/0)

0/6

5/5

8/4

9/3

8/2

5/1

(0/0)

c

c′

g′

c′′

e′′

etc.

Minor

2/0

1/1

(0/0)

3/0

2/2

1/2

(0/0)

4/0

3/3

2/4

1/3

(0/0)

5/0

4/4

3/6

2/6

1/4

(0/0)

6/0

5/5

4/8

3/9

2/8

1/5

(0/0)

c

c,

f,,

c,,

as,,,

etc.

Figure 169

Here
is the proof that they are parabolas: the familiar parabola equation x2 = 2px changes into y2
= x for a parabola whose parameter is
1/2, i.e. the y-coordinates
(perpendicular lines) are equal to the square roots of the corresponding x-coordinates (parallel lines). For the
parabola 0/65/58/49/38/25/10/0
in Fig. 168, 5/1 is the perpendicular line 2 units away
from the point 5/3 on the x-axis, and the length of the x-line
9/3-5/3 contains 4 units. The y-value 2, then, is the square root of
the x-value 4. The x-value 0/3-9/3
has 9 units as its axis, the corresponding y-value
9/0-9/3 = 3 units. √9 = 3,
etc. The apexes of these parabolas generate further parabolas. We obtain a
beautiful image of these parabolas (Fig. 170) from their fourfold combination,
anticipating what will be further discussed in §32.

The
hyperbola also has a simple and interesting harmonic derivation. If we draw the
partial-tone-values of its string-length measures perpendicularly (Fig. 171)
and turn them sideways, always using unity as a measure, then we get perfect
rectangles, identical in area to the unit-square. Connecting the corners then produces
a hyperbola, whose equation is a2
= xy, as is generally known. In our
case, this means that

1/1 · 1/1

1/2 · 2/1

1/3 · 3/1

etc.

}

= 1

Figure 170

Figure 171

Figure 172

As we saw above, the hyperbola is
the geometric location for all points for which the difference between the x- and y-coordinates is the same. Thus we can also explain their
“harmonics,” as in Fig. 172.

The
hyperbola, drawn in points, continuing endlessly in both the x- and y-directions, indicates that from any point placed on it, a
rectangle of consistently equal area can be introduced between the curve and
the axes A B C. If d – B = 1, then we have:

for:

length:

height:

therefore, the
quadrilateral’s area:

a2

1/4

4/1

1/4
· 4/1 = 1

b2

1/2

2/1

1/2
· 2/1 = 1

c2

3/4

4/3

3/4
· 4/3 = 1

d2

1

1

1 · 1 = 1

e2

4/3

3/4

4/3
· 3/4 = 1

f2

2/1

1/2

2/1
· 1/2 = 1

g2

4/1

1/4

4/1
· 1/4 = 1

Figure 173

The
law of hyperbola construction therefore shows us an increasing arithmetic
series (1/n2/n3/n4/n ...) and a decreasing geometric series (harmonic n/1n/2n/3 ...)—a precise analogy to
the intersecting major-minor series of our diagram.

And
if we consider, moreover, that the ellipse is the geometric location for all
points for which the sum of two distances has an unchanging value, then it is
easy enough to construct the ellipse harmonically with reciprocal partial-tone
logarithms, since their sum is always 1—for example, 585 g (3/2) + 415 f (2/3) = 1000. In Fig. 174, this tone-pair
is drawn with a thick line and marked for clarification. We mark two focal
points 8 cm apart (Fig. 174) for the construction of the ellipse, draw one
circle around one focal point at radius 5.9 cm (585 g) and one around the other at radius 4.1 cm (415 f), then trace the intersection points
of each pair of rays, up to the point where the two shorter f-rays intersect with the circumference
of a small circle drawn around the center of the ellipse, and the two longer g-rays intersect with the circumference
of a larger circle drawn around the center of the ellipse. These two outer circles,
whose radii are of arbitrarily length, serve simply to intercept the vectors
(directions) of the single tone-values and to distinguish them clearly from one
another. All other points of the ellipse are constructed in the same way. The
tone-logarithms here were simply chosen in order for the construction of the ellipse
points to be as uniform as possible. If the reader has a good set of drawing
instruments, then he can use all of index 16 for point-construction—a beautiful
and extremely interesting project. In this case it would be best to use focal
points 16 cm apart, and to double the logarithmic numbers.

Figure 174

Even
if this construction of an ellipse from the equal sums of focal-point rays is
nothing new and can be found in every elementary textbook, its construction
from the reciprocal P-logarithms still gives us an important new realization.
As one can see from the opposing direction of rays in the ratio progression of
the outer and inner circles, the tones are arranged here in each pair of
octave-reduced semicircles, and thus the directions of the ratios of the two
circles are opposite to each other. From the viewpoint of akróasis, then, there are two polar directions of values
concealed in the ellipse: a result that might alone justify harmonic
analysis as a new addition to a deeper grasp of the nature of the ellipse.

Parabola,
hyperbola, ellipse, and circle (in §33 we will discuss the harmonics of
circular arrangements of the P) are of course conic sections, i.e. all these
figures can be produced from certain sections of a cone, or of two cones
tangent at their apexes. The above harmonic analyses, of which many more could
be given, show that these conic sections are closely linked to the laws of
tone-development, which supports the significance of the cone as a
morphological prototype for our point of view. In pure mathematics, this significance
has been known since Apollonius, renewed by Pascal, and discussed in De la
Hire’s famous work Sectiones Conicae,
1585 (the reader should definitely seek out a copy of this beautiful volume at
a library), right up to modern analytical and projective geometry. For those
interested in geometric things and viewpoints, hardly anything is more
wonderful than seeing the figures of these conic sections emerge from an almost
arbitrary projection of points and lines, aided only by a ruler. For a
practical introduction see also L. Locher-Ernst’s work, cited in §24c.

§27a. Ektypics

Mathematically
speaking, ellipses, parabolas, and hyperbolas can be defined as the geometric
location of all points for which the distance from a fixed point (the focal
point) is in a constant relationship to the distance from a fixed straight line
(the directrix). On this rest the projective qualities of conic sections and
the possibility of constructing them by means of simple straight lines (the
ruler).

In
detail, as remarked above, these “curves of two straight lines” have many more
specific harmonic attributes—for example, the octave relationship (1 : 2) of
the areas of a rectangle divided by a parabola, the graphic representation of
harmonic divisions in the form of hyperbolas, etc. One obtains the “natural
logarithm” when one applies the surface-content enclosed by the hyperbola
between the two coordinates (F. Klein: Elementarmathematik
vom höheren Standpunkt aus, 1924, p. 161); thus, a close relationship also
exists between the conic sections and the nature of the logarithm. The
applications of the laws of the conic section are many, especially in the exact
natural sciences. I will mention only the Boyle-Marriott Law, which connects
the respective number-values of pressure and volume, and in which the hyperbola
emerges as a graphic expression (and thus the pressure : volume ratio of the
reciprocal partial-tone values 4 : 1/4, 2 : 1/2,
1 : 1, 1/2 : 2, etc. are expressed most beautifully). I
am also reminded of the “parabolic” casting curves in mechanics, the properties
of focal points, parabolas in optics, the countless “asymptotic” relationships,
etc. Admittedly, these applications are mostly obscured by differential and
integral calculus, though doubtless simplified mathematically—in other words,
the morphological content of conic sections is outwardly diminished in favor of
a practical calculation method, but remains the same in content.

Figure 175a

Figure 175b

Because
of this, it is not astonishing when a figure such as a cone, from which all
these laws flow as from the source of an almost inexhaustible spring of forms,
is applied emblematically even in the most recent deliberations of natural
philosophy, as a direct prototype for the “layers of the world” and for our “causal
structure.” In Figures 175a and b I reproduce the diagrams from H. Weyl: Philosophie der Mathematik und
Naturwissenschaft, 1927, pp. 65 and 71, which speak for themselves.